I have an image at the background and so sometimes it would be better if I could move the title a bit to make it more readable against the background. I understand I can move the title itself by adding \vspace in the {format} section of \titleformat. Problem is that this doesn't move the number of the chapter.
All my chapter headers are at the top left of the page, I believe because of this code in the *cls file
The \titleformat command is provided by the titlesec package. You might want to look at its documentation.
It's a bit hard to tell, since you didn't say what document class you are using, nor did you provide a MWE per board rules, but using titlesec it's usually possible to prevent chapters from beginning on new pages (and override this setting in the documentclass) by its \titleclass command, and then defining the \chapterbreak to be empty:
It also provides a \titlespacing command you can use to customize various aspects of the spacing of the titles. Without more information about what you want to do, I can't really help with that. However, maybe the documentation I linked above would provide some guidance.
However, if this is your thesis, if I were you, I'd find out from your University what kind of formatting is allowed. Most university have very narrow guidelines about how theses are formatted, and deviating from them is not allowed.
I did try your mwe above with a couple of my chapters but got an error related to the \lipsum command, certainly because I don't know how to adapt it to my thesis but I couldn't find any documentation to make me understand what it actually does. Besides, I don't know if I can use it with my chapters being split between different documents.